Module optimization_engine::constraints

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Constraints and projections

This module defines the trait Constraint, which specifies an abstract projection method, and a collection of simple sets, such as norm-balls, finite sets, second-order cones and their Cartesian products.

Structs§

  • AffineSpace
    An affine space here is defined as the set of solutions of a linear equation, $Ax = b$, that is, $E=\{x\in\mathbb{R}^n: Ax = b\}$, which is an affine space. It is assumed that the matrix $AA^\intercal$ is full-rank.
  • Ball1
    A norm-1 ball, that is, a set given by $B_1^r = \{x \in \mathbb{R}^n {}:{} \Vert{}x{}\Vert_1 \leq r\}$ or a ball-1 centered at a point $x_c$, that is, $B_1^{x_c, r} = \{x \in \mathbb{R}^n {}:{} \Vert{}x-x_c{}\Vert_1 \leq r\}$
  • Ball2
    A Euclidean ball, that is, a set given by $B_2^r = \{x \in \mathbb{R}^n {}:{} \Vert{}x{}\Vert \leq r\}$ or a Euclidean ball centered at a point $x_c$, that is, $B_2^{x_c, r} = \{x \in \mathbb{R}^n {}:{} \Vert{}x-x_c{}\Vert \leq r\}$
  • BallInf
    An infinity ball defined as $B_\infty^r = \{x\in\mathbb{R}^n {}:{} \Vert{}x{}\Vert_{\infty} \leq r\}$, where $\Vert{}\cdot{}\Vert_{\infty}$ is the infinity norm. The infinity ball centered at a point $x_c$ is defined as $B_\infty^{x_c,r} = \{x\in\mathbb{R}^n {}:{} \Vert{}x-x_c{}\Vert_{\infty} \leq r\}$.
  • CartesianProduct
    Cartesian product of constraints
  • EpigraphSquaredNorm
    The epigraph of the squared Eucliden norm is a set of the form $X = \{x = (z, t) \in \mathbb{R}^{n}\times \mathbb{R} {}:{} \|z\|^2 \leq t \}.$
  • FiniteSet
    A finite set, $X = \{x_1, x_2, \ldots, x_n\}\subseteq\mathbb{R}^n$, given vectors $x_i\in\mathbb{R}^n$
  • Halfspace
    A halfspace is a set given by $H = \{x \in \mathbb{R}^n {}:{} \langle c, x\rangle \leq b\}$.
  • Hyperplane
    A hyperplane is a set given by $H = \{x \in \mathbb{R}^n {}:{} \langle c, x\rangle = b\}$.
  • NoConstraints
    The whole space, no constraints
  • Rectangle
    A rectangle, $R = \{x \in \mathbb{R}^n {}:{} x_{\min} {}\leq{} x {}\leq{} x_{\max}\}$
  • SecondOrderCone
    A second-order cone (SOC)
  • Simplex
    A simplex with level $\alpha$ is a set of the form $\Delta_\alpha^n = \{x \in \mathbb{R}^n {}:{} x \geq 0, \sum_i x_i = \alpha\}$, where $\alpha$ is a positive constant.
  • Sphere2
    A Euclidean sphere, that is, a set given by $S_2^r = \{x \in \mathbb{R}^n {}:{} \Vert{}x{}\Vert = r\}$ or a Euclidean sphere centered at a point $x_c$, that is, $S_2^{x_c, r} = \{x \in \mathbb{R}^n {}:{} \Vert{}x-x_c{}\Vert = r\}$
  • Zero
    Set Zero, $\{0\}$

Traits§